Degenerate Critical Points and Catastrophes: Fold Catastrophe

The simple algebraic curve is a good enough example to explain degeneracy and catastrophe in the extended phase space . With the help of this Demonstration, students can easily understand the fold catastrophe.

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Bifurcation-catastrophe theorists roughly define a catastrophe as a sudden transition resulting from a continuous parameter change. Here are some basic definitions for understanding the fold catastrophe.

1. A critical point of a differentiable function of one variable satisfies .

2. A nondegenerate critical point of a differentiable function of one variable satisfies and ; if and , is called a degenerate critical point.

For and , there are two nondegenerate critical points; for , there is one degenerate critical point; and for , there are no critical points.